IN THIS PAPER, WITH A NEW SIMPLE PROOF, AN IMPORTANT INEQUALITY FOR A CONTRACTION INTEGRAL EQUATION IS OBTAINED. FROM A PRACTICAL PROGRAMMING POINT OF VIEW, THIS INEQUALITY ALLOWS TO EXPRESS ITERATIVE ALGORITHM WITH A ”FOR LOOP” RATHER THAN A ”WHILE LOOP”. THE MAIN TOOL USED IN THIS PAPER IS THE FIXED POINT THEOREM IN THE LEBESGUE SPACE.